Gödel’s Theorem’s Philosophical Implications

I’m a few days late to this party, but Ali Eteraz writes about the philosophical implications of Gödel’s Incompleteness Theorem. I’m going to have to disagree with Ali’s entire post here, because Gödel’s work is only relevant to a very limited portion of mathematics.

Although I have to face the possibility that some statements are undecidable, in practice very few important conjectures turn out to be undecidable. The most famous conjectures are at least partially decidable, in the sense that if they’re wrong, we can always prove it. If Fermat’s last theorem had been wrong, then it would’ve been possible to show it by a brute force computation given enough computer-hours.

Undecidable propositions tend to crop up in places where we can’t even define things explicitly. The undecidable continuum hypothesis roughly states that given any infinite subset of R, S, there exists a bijective function from S to either Z or R; but in most cases we can’t even define that function explicitly – even for something as simple as S = [0, 1] = {x in R: 0 <= x <= 1} we run into problems.

For all practical intents, we can ignore undecidable propositions, then, just like we can ignore the fact that many decidable computations have running times longer than the age of the Universe.

The other thing Gödel proved, that we can’t prove mathematics consistent, is even less relevant. Yes, it means mathematics rests on the assumption that a certain basal axiomatic system is consistent, which is a scientific rather than rigorous assumption. So what? Newton developed mechanics without rigorously proving that gravity existed.

Proving that a system is consistent is again partially decidable. If an axiomatic system is inconsistent, it’s possible to prove it – in fact, there’s a theorem that says a system is inconsistent iff it can prove its own consistency (no typo). The fact that mathematics has stood unproven inconsistent for a few thousand years, even though the construction of an inconsistency would be immediately noticeable in everyday mathematics, suggests that we can assume mathematics is consistent until proven otherwise.

None of this demonstrates limits to reason. Gödel’s Theorem proves in an almost trivial way that reason is limited, but it doesn’t provide an alternative. At the limits of the most abstract mathematical thought, reason fails; however, any commonly touted alternative – faith, relativism, tradition, our own beliefs about how things ought to be – fails routinely and spectactularly. And at any rate, no alternative can ever prove mathematically undecidable propositions or establish the consistency of mathematics, so any limit to reason is a limit to everything.

One Response to Gödel’s Theorem’s Philosophical Implications

  1. Brian Coyle says:

    Eight years later, I came across this. Perhaps the author has disappeared. Still, I think something has been missed. Godel’s proof was really a disproof. He showed what could not be asserted. This was, indeed, all the he could do, because, like the system he analyzed, his analysis was on the same plane as what he analyzed. I use the word plane descriptively, but not without reason. The Romans codified Aristotle’s logic in the word explanation, meaning ex-planus, or apart from the plane. Achieving what today we call theory requires a meta-level perspective. But math is unique among sciences, because its reality isn’t concrete. We use the terms inference and evaluation just as physical and social sciences do, but in reference to ideas, not objects, organisms, or processes. Einstein could explain gravity by discovering a new dimension to perceive from. Darwin’s meta-level was also a new way to view time. Explanation that leads to deep theory has this component, which challenges the mathematician. It’s fair to say Stephen Wolfram’s algorithms, and indeed the IT mathematics of Group, Category, and Proof theory, may be meta-level explanations. Set theory wasn’t, which is why Godel matters. Did that stop set theory’s use? No, but it limits is wider applicability.

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